Physics equations/Equations/Rotational and linear motion analogy

The following table refers to rotation of a rigid body about a fixed axis: $𝐬$ is arclength, $𝐫$ is the distance from the axis to any point, and ${𝐚}_{𝐭}$ is the tangential acceleration, which is the component of the acceleration that is parallel to the motion. In contrast, the centripetal acceleration, ${𝐚}_{𝐜}=v^2/r={\omega }^{2}r$, is perpendicular to the motion. The component of the force parallel to the motion, or equivalently, perpendicular, to the line connecting the point of application to the axis is ${𝐅}_{\perp }$. The sum is over $𝐣=1𝐭𝐨N$ particles or points of application.

Analogy between Linear Motion and Rotational motion

Linear motion	Rotational motion	Defining equation
Displacement = $𝐱$	Angular displacement = $\theta$	$\theta =𝐱/𝐫$
Velocity = $𝐯$	Angular velocity = $\omega$	$\omega =𝐝\theta /𝐝𝐭=𝐯/𝐫$
Acceleration = $𝐚$	Angular acceleration = $\alpha$	$\alpha =𝐝\omega /𝐝𝐭={𝐚}_{𝐭}/𝐫$
Mass = $𝐦$	Moment of Inertia = $𝐈$	$𝐈=\sum {𝐦}_{𝐣}{{𝐫}_{𝐣}}^2$
Force = $𝐅=𝐦𝐚$	Torque = $\tau =𝐈\alpha$	$\tau =\sum {𝐫}_{𝐣}{𝐅}_{\perp }{}_{𝐣}=\sum {𝐫}_{\mathbf{\perp }𝐣}{𝐅}_{{}_{𝐣}}$
Momentum= $𝐩=𝐦𝐯$	Angular momentum= $𝐋=𝐈\omega$	$𝐋=\sum {𝐫}_{𝐣}𝐩{}_{𝐣}$
Kinetic energy = $\frac{1}{2}𝐦{𝐯}^2$	Kinetic energy = $\frac{1}{2}𝐈{\omega }^2$	$\frac{1}{2}\sum {𝐦}_{𝐣}{{𝐯}_{𝐣}}^2=\frac{1}{2}\sum {𝐦}_{𝐣}{{𝐫}_{𝐣}}^2{\omega }^2$